Brachistochrone curve

In mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brakhistos khrónos), meaning "shortest time"),[1] or curve of fastest descent, is the curve that would carry an idealized point-like body, starting at rest and moving along the curve, without friction, under constant gravity, to a given end point in the shortest time. For a given starting point, the brachistochrone curve is the same as the tautochrone curve.

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Given two points A and B, with A not lower than B, only one upside down cycloid passes through both points, has a vertical tangent line at A, and has no maximum points between A and B: the brachistochrone curve. The curve does not depend on the body's mass or on the strength of the gravitational constant.

According to Fermat’s principle: The actual path between two points taken by a beam of light is the one which is traversed in the least time. In 1697 Johann Bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g).[3]

Johann's brother Jakob showed how 2nd differentials can be used to obtain the condition for least time. A modernized version of the proof is as follows. If we make a negligible deviation from the path of least time, then, for the differential triangle formed by the displacement along the path and the horizontal and vertical displacements,

.

On differentiation with dy fixed we get,

.

And finally rearranging terms gives,

where the last part is the displacement for given change in time for 2nd differentials. Now consider the changes along the two neighboring paths in the figure below for which the horizontal separation between paths along the central line is d2x (the same for both the upper and lower differential triangles). Along the old and new paths, the parts that differ are,

For the path of least times these times are equal so for their difference we get,

Johann Bernoulli posed the problem of the brachistochrone to the readers of Acta Eruditorum in June, 1696. He published his solution in the journal in May of the following year, and noted that the solution is the same curve as Huygens's tautochrone curve. After deriving the differential equation for the curve by the method given above, he went on to show that it does yield a cycloid.[4][5] However, his proof is marred by his use of a single constant instead of the three constants, vm, 2g and D, above. Five mathematicians responded with solutions: Isaac Newton, Jakob Bernoulli (Johann's brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital. Four of the solutions (excluding l'Hôpital's) were published in the same edition of the journal as Johann Bernoulli's. In his paper Jakob Bernoulli gave a proof of the condition for least time similar to that above before showing that its solution is a cycloid.[4]According to Newtonian scholar Tom Whiteside, Newton found the problem in his mail, in a letter from Johann Bernoulli,[6] when he arrived home from the mint at 4 p.m., and stayed up all night to solve it and mailed the solution by the next post. This story gives some idea of Newton's power, since Johann Bernoulli took two weeks to solve it.[7]

Earlier, in 1638, Galileo had tried to solve a similar problem for the path of the fastest descent from a point to a wall in his Two New Sciences. He draws the conclusion (Third Day, Theorem 22, Prop. 36) that the arc of a circle is faster than any number of its chords,[8]

"From the preceding it is possible to infer that the quickest path of all [lationem omnium velocissimam], from one point to another, is not the shortest path, namely, a straight line, but the arc of a circle.

...

Consequently the nearer the inscribed polygon approaches a circle the shorter is the time required for descent from A to C. What has been proven for the quadrant holds true also for smaller arcs; the reasoning is the same."

We are warned earlier in the Two New Sciences (just after Theorem 6) of possible fallacies and the need for a "higher science." In this dialogue Galileo reviews his own work. The actual solution to Galileo's problem is half a cycloid. Galileo studied the cycloid and gave it its name, but the connection between it and his problem had to wait for advances in mathematics.